On the Long Time Existence of a Fractional KdV-BBM Type Equation

We consider a fractional Korteweg de Vries-Benjamin Bona Mahony (KdV-BBM) type equation including both fractional dispersive terms of fractional KdV and fractional BBM equations. We aim to enhance the existence time of solutions with small initial data $\mathrm{\mid }\mathrm{\mid }{u}_0\mathrm{\mid }{\mathrm{\mid }}_{H^{N+\alpha \mathrm{/}2}}=\epsilon$ from $\frac{1}{\epsilon }$ to $\frac{1}{{\epsilon }^2}$. The proof relies on the combination of a modified energy method with Fourier techniques. In addition, the long time existence issues are investigated numerically. Numerical observations of the lifespan give an evidence of existence of solutions beyond the hyperbolic time scale. This study provides a detailed analysis from both analytical and numerical aspects for the existence of smooth solutions.